The law has been on the books for ages (I remember buying a license as a kid), but has never been seriously enforced. It seems to have originated as a way to help trace stolen bikes (though the sticker is easy to remove), but lately the LAPD has been using it as an excuse to harass cyclists--and we're not talking Lycra-clad Richie Riches on Cervelos, of course, but cyclists seen as political or countercultural, such a fixie punx, Midnight Ridazz, CMers, etc. (Yours truly, though an anarcho-syndicalist, has a white beard on his chin and fenders on his fixie, and so gets friendly nods from cops....)
To add injury to insult, LAPD assessed fines of up to $200 for an "infraction," though the regulation calls for a $10 fine at worst. And furthermore, though police stations (as well as bike shops) are supposed to sell the licenses, almost no police stations carry the licenses, and those that do sell them only a couple of hours a week.
Obviously a rotten rule. Even the Los Angeles Business Journal this week published an article scorning the law and its effect on the bike business in LA.
Read all about it on Streetsblog, which should also carry followup news on the matter.